Essays on derivatives pricing in incomplete financial markets
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This dissertation is a contribution to the valuation and risk management of derivative securities in incomplete financial markets. It consists of two parts dedicated to two distinct valuation methodologies. In the first part, we develop a valuation approach based on equilibrium arguments from the perspective of option market makers and financial intermediaries. This approach produces a new pricing concept that we call the competition-based price. We analyze such prices in both a semimartingale and a diffusion setting. The emerging pricing measure is characterized as the minimal entropy martingale measure (MEMM) with respect to a new prior. This new prior depends on the aggregate demand and inventory of the derivatives and is characterized as an Esscher transform of the historical measure. In a diffusion setting, the pricing measure is explicitly constructed. We show that the competitive price of a derivative is an increasing function of the demand of any derivative in the market. The increasing rate is proportional to the covariance between the unhedgeable parts of the associated derivative payoffs, calculated under the competition-based pricing measure. This result may contribute to the resolution of some of the well known option-pricing puzzles. We further compare our approach to existing pricing methodologies, such as the marginal-utility pricing and indifference valuation. In addition, we apply our approach to price a family of volatility derivatives. We develop numerical schemes based on Monte Carlo simulations for a Heston-type stochastic volatility model. In the second part, we apply the well established indifference approach to value options with staging structure and sequential decisions, such as installment options and venture capital contracts. In a diffusion market setting, we analyze the underlying stochastic optimization problems via the associated Hamilton-JacobiBellman equations. We deduce a quasilinear PDE for the indifference price and analyze it probabilistically. We also obtain an explicit pricing formula under appropriate market restrictions and characterize the indifference price as a nonlinear expectation under the MEMM. The associated hedging and risk monitoring strategies are investigated. We further develop numerical schemes based on regression techniques to value the ASX Installments and the staged financing of venture capital. Moreover, a foresighted valuation framework is introduced to incorporate the investors’ private information into their valuation and hedging strategies. Such information may include both their ex-ante risk exposure and ex-post investment opportunities. Finally, we adopt the recently developed dynamic performance criteria to price volatility derivatives. We develop numerical schemes for the computation of the forward and backward indifference prices in models of Heston and reciprocalHeston type.